Understanding Compactness: The Harmony of Boundedness, Closedness, and Convergence in Real Analysis

One of the distinguishing properties of a bounded closed interval [a, b] is that every sequence in it has a subsequence converging to a limit in the interval. This need not happen with an unbounded interval such as [0, 1) or a bounded non closed interval such as (0,1]; the former contains the sequence {n}n≥1, which has no convergent subsequence, and the latter contains the sequence {1/n} n≥1, which has no subsequence converging to a limit belonging to the interval. In fact, it is true of any bounded closed subset of R that any sequence in it has a subsequence converging to a limit belonging to the subset. To see why, we first note that any sequence in a bounded subset must, by the Bolzano-Weierstrass theorem have a convergent subsequence with limit in R; this limit must then be in the closed subset by the definition of a closed subset. A compact set in R is a set E satisfying the property that if U is a collection of open sets in R whose union contains E, then there is a finite sub collection V of U whose union contains E. Recall that such a collection is called an open cover and V is called a finite sub cover of U for E. In terms of this, a set E in R is compact if every open cover of the set E has a finite sub cover for E. Because of these criteria, compact sets are also viewed as a generalization of finite sets.